(* Catalan Dual of Λ((111) Great Rhombicosidodecahedron (Disdyakis Triacontahedron) *)

Please note : My default Mathematica notebooks unfortunately have the symbol φ being the small GoldenRatio (.618) vs . 1.618 . 
    I don' t use the symbol ϕ as in the article for the Disdyakis Triacontahedron .
    When creating output for WP I convert φ->1/φ to be consistent (e.g. as shown in the WP R scale factor(s) section below)

Set up quaternion H3 Coxeter-Dynkin Weyl Orbits W(H3) (Koca)

H3oSet:=((* Set up for doW(in) with
 α1,α2,α3,α4 *)
α1=(Subscript[ℯ, 1]);
α2=(1/2)(τ (Subscript[ℯ, 1]) +(Subscript[ℯ, 2])+σ (Subscript[ℯ, 3]));
α3=(Subscript[ℯ, 2]);
(* r1, r2,r3,r4 sequences *)
aStr="r1@r2@r3";
bStr="r2@r3";
cStr="r2@r3@r2";
(* the (Cartan) Schlafli Matrix *)
csm=csmH3={{2,-τ,0},{-τ,2,-1},{0,-1,2}}/.slRep;
(* {α0,β0,γ0,ϱ0} which is used in Λ1[[aa1,aa2,aa3,aa4] which drives xxOct[x1,x2,x3,x4] *)
α0=-aa1/2;
β0=-aa3/2;
γ0=-τ(τ aa1+2 aa2+aa3)/2;ϱ0=0;);

doW@"H3"

Koca quaternion coordinate scales (See section 4.5 of  https://arxiv.org/abs/0908.3272)

λ1=((τ+3)/5)/.slRep;
η1=((2σ+3)/3)/.slRep;

(* ABC triangle vertices as decagonal,hexagonal and the square great rhombicosidodecahedron face centers *)
A1=λ1(-σ (Subscript[ℯ, 2])+(Subscript[ℯ, 3]))/.slRep;
B1=η1(-σ (Subscript[ℯ, 1])+τ (Subscript[ℯ, 3]))/.slRep;
C1=( (Subscript[ℯ, 1])-σ (Subscript[ℯ, 2])+τ (Subscript[ℯ, 3]))/2/.slRep;

(* q is a point orthogonal to ABC *)
q1=( τ (Subscript[ℯ, 1])-2σ (Subscript[ℯ, 2])+(τ^3) (Subscript[ℯ, 3]))/2/.slRep;

(* q' from q rotated by π/5 around A, 
q' is orthogonal to AB'C', 
where B' & C' are rotated by π/5 around A *)
q1p=( 2(Subscript[ℯ, 1])+(Sqrt[5]) (Subscript[ℯ, 2])+(τ+2) (Subscript[ℯ, 3]))/2/.slRep;

(* circumradius of the Icosahedron *)
aNorm=FullSimplify@octNorm@A1
N[%/.φωRep]
1/%

(* Icosahedron scaling *)
iNorm=1/FullSimplify[Norm@{0,1,1/φ}/.slRep,Assumptions->φωAssumptions]
N[%/.φωRep]

(* circumradius of the Dodecahedron *)
bNorm=FullSimplify@octNorm@B1
N[%/.φωRep]
1/%

(* Dodecahedron scaling *)
oNorm=1/FullSimplify@Norm@{1,1,1}
N@%;

(* circumradii of the Icosidodecahedron *)
cNorm=FullSimplify@octNorm@C1
N[%/.φωRep]

Original (incorrectly trying to correct Koca & MTM coordinates  using WP R) Visualization of the Disdyakis Triacontahedron 
     with Icosahedron circumradius=1, (Sqrt[3])iNorm=.9106 scaled Dodecahedron, S=aNorm=.92101 scaled Icosadodecahedron

DisdyakisTriacontahedronKoca=FullSimplify[Join[{
(* 62=32+30=12+20+30 Norms *)
(* Icosahedron (12) **)
{"H3",2iNorm{0,0,1}},

(* Dodecahedron (20=Icosahedron (12) + Cube (8)) **)
(* Icosahedron (12) *)
{"H3",2iNorm{σ,0,0}},
(* Cube (8) *)
{"H3",2iNorm{1,-1,1},"None"},

(* Icosidodecahedron (30=Octahedron (6)+Rhombicuboctahedron (24)) **)
(* Octahedron (6) **)
{"H3",(1/aNorm) {0,σ,0}},
(* Rhombicuboctahedron (24) **)
{"H3",(1/aNorm){-1,τ,σ}}}]/.slRep,Assumptions->φωAssumptions]
dtVertexListKoca=oScaleListDo@%;
checkVertices[%,True,True,True,True,False,False]
opacity=.5;
hulls3D@%%%
dtPolyKoca=overallHull;

3D Polyhedron angles  for the ABC surface triangle AB, BC, AC

FullSimplify[A1⊕B1]
FullSimplify[%/.φωRep]
N@%
ArcCos[%]/°
180-%
DMSList@%

FullSimplify[A1⊕C1]
FullSimplify[%/.φωRep]
N@%
ArcCos[%]/°
180-%
DMSList@%

FullSimplify[B1⊕C1]
FullSimplify[%/.φωRep]
N@%
ArcCos[%]/°
180-%
DMSList@%

2D Polygon angles for the ABC surface triangle for dtVertexListKoca

(* Project the face triangle to the XY plane *)
Off[Nearest::mepcmp]
(* Get the index to the 3 vertices A,B,C *)
abc3D=Flatten[FullSimplify[Nearest[dtVertexListKoca/.φωRep,#]&/@(oct2ImQuat@#&/@{A1,B1,C1}/.φωRep)],1];
%//MatrixForm
abc3Dindex=position[rndMat[dtVertexListKoca/.φωRep],#]&/@rndMat@abc3D

(* Define a 3D Polygon from the ABC vertices *)
abc3DVerts=FullSimplify[dtVertexListKoca[[abc3Dindex]],Assumptions->φωAssumptions];
%//MatrixForm
%/.φωRep//MatrixForm
N@%//MatrixForm

(* Get the 3D edge lengths for AB, AC, BC *)
FullSimplify[Norm@{abc3DVerts[[#[[1]]]]-abc3DVerts[[#[[2]]]]}/.slRep,Assumptions->φωAssumptions]&/@Subsets[Range@3,{2}];
%/.φωRep//MatrixForm
N@%//MatrixForm

(* Define a MTM Polygon *)
abcPoly=Polygon@abc3DVerts;

(* Get angle A *)
aAngle=PolygonAngle[abcPoly/.φωRep,abc3DVerts[[1]]/.φωRep];
N[%/°/.slRep]
DMSList@%

(* Get angle B *)
bAngle=PolygonAngle[abcPoly/.φωRep,abc3DVerts[[2]]/.φωRep];
N[%/°/.slRep]
DMSList@%

(* Get angle C *)
cAngle=PolygonAngle[abcPoly/.φωRep,abc3DVerts[[3]]/.φωRep];
N[%/°/.slRep]
DMSList@%

abcAngles={aAngle,bAngle,cAngle};
abcAnglesN=N[abcAngles/°/.φωRep/.slRep]
Total@%

MTM Disdyakis Triacontahedron coordinates  & visualization (uses scaled up and rotated original Koca visualization coordinates)

(Eric Weisstein Mathworld  https://mathworld.wolfram.com/DisdyakisTriacontahedron.html)

mtmVerts=PolyhedronData["DisdyakisTriacontahedron","VertexCoordinates"];
mtmNorms=Sort[Union[FullSimplify@Norm@#&/@%],N@#1>#2&]
N@%

hulls3D@mtmVerts

hulls3D[(mtmVerts/(mtmNorms[[1]]))]

mtmNorm21=FullSimplify[mtmNorms[[2]]/mtmNorms[[1]]]
N@%

mtmNorm31=FullSimplify[mtmNorms[[3]]/mtmNorms[[1]]]
N@%

mtmNorm32=FullSimplify[mtmNorms[[3]]/mtmNorms[[2]]]
N@%

MTM Norms are precisely the same as the Koca Norms

(* Compare Koca's circumradius of the Icosahedron to Dodecahedron *)
idCompare=FullSimplify[bNorm/aNorm,Assumptions->φωAssumptions]
N[%/.φωRep]
FullSimplify@mtmNorm21==FullSimplify[idCompare/.φωRep]

(* Compare Koca's circumradius of the Icosidodecahedron to Dodecahedron *)
iddCompare=FullSimplify[cNorm/bNorm,Assumptions->φωAssumptions]
N[%/.φωRep]
FullSimplify@dtNorm32==FullSimplify[iddCompare/.φωRep]

Koca Dihedral angle and AB, BC, AC angles from origin

Graphics3D[{(*)Opacity@.6,grPoly,**)
Opacity@.6,dtPolyKoca,
Opacity@1,PointSize[.02],
(* Primary reference point A *)
Black,{Point@#,Line[{{0,0,0},#}]}&@N[((oct2ImQuat@A1)/bNorm)/.φωRep/.slRep],
(* Primary reference point B *)
Red,{Point@#,Line[{{0,0,0},#}]}&@N[((oct2ImQuat@B1)/bNorm)/.φωRep/.slRep],
(* Primary reference point C *)
Green,{Point@#,Line[{{0,0,0},#}]}&@N[((oct2ImQuat@C1)/bNorm)/.φωRep/.slRep],
(* Orthogonal reference point orthogonal to primary triangle face q *)
Blue,{Point@#,Line[{{0,0,0},#}]}&@N[((oct2ImQuat@q1)/2)/.φωRep],
(* Dihedral angle via the π/5 rotation of q to q' *)
Magenta,{Point@#,Line[{{0,0,0},#}]}&@N[((oct2ImQuat@q1p)/2)/.φωRep],
Orange,{Point@#,Line[{{0,0,0},#}]}&/@N[abc3DVerts/.φωRep]}]

(* Dihedral angle via the π/5 rotation of q to q' *)
FullSimplify[q1⊕q1p]
FullSimplify[%/.φωRep]
N@%
ArcCos[%]/°
180-%
DMSList@%
θ=%%°

Visualization of  the (spherical single hull form) Disdyakis Triacontahedron with circumradius=1

DisdyakisTriacontahedronSphere=FullSimplify[Join[{
(* 62=32+30=12+20+30 Norms *)

(* Icosahedron (12) **)
{"H3",2iNorm{0,0,1}},

(* Dodecahedron (20=Icosahedron (12) + Cube (8)) **)
(* Icosahedron (12) *)
{"H3",2oNorm{σ,0,0}}/.slRep,
(* Cube (8) *)
{"H3",2oNorm{1,-1,1},"None"},

(* Icosidodecahedron (30=Octahedron (6)+Rhombicuboctahedron (24)) **)
(* Octahedron (6) **)
{"H3",{0,σ,0}},
(* Rhombicuboctahedron (24) **)
{"H3",{-1,τ,σ}}}]/.slRep,Assumptions->φωAssumptions]
dtSphereVerts=oScaleListDo@%;
checkVertices[%,True,True,True,True,False,False]
opacity=.6;
hulls3D@%%%
dtSpherePoly=overallHull;

2D Polygon angles for the ABC surface triangle for dtSphereVerts

(* Project the face triangle to the XY plane *)
Off[Nearest::mepcmp]
(* Get the index to the 3 vertices A,B,C *)
abc3D=Flatten[FullSimplify[Nearest[dtSphereVerts/.φωRep,#]&/@(oct2ImQuat@#&/@{A1,B1,C1}/.φωRep)],1];
%//MatrixForm
abc3Dindex=position[rndMat[dtSphereVerts/.φωRep],#]&/@rndMat@abc3D

(* Define a 3D Polygon from the ABC vertices *)
abc3DVerts=FullSimplify[dtSphereVerts[[abc3Dindex]],Assumptions->φωAssumptions];
%//MatrixForm
%/.φωRep//MatrixForm
N@%//MatrixForm

(* Get the 3D edge lengths for AB, AC, BC *)
FullSimplify[Norm@{abc3DVerts[[#[[1]]]]-abc3DVerts[[#[[2]]]]}/.slRep,Assumptions->φωAssumptions]&/@Subsets[Range@3,{2}];
%/.φωRep//MatrixForm
N@%//MatrixForm

(* Define a MTM Polygon *)
abcPoly=Polygon@abc3DVerts;

(* Get angle A *)
aAngle=PolygonAngle[abcPoly/.φωRep,abc3DVerts[[1]]/.φωRep];
N[%/°/.slRep]
DMSList@%

(* Get angle B *)
bAngle=PolygonAngle[abcPoly/.φωRep,abc3DVerts[[2]]/.φωRep];
N[%/°/.slRep]
DMSList@%

(* Get angle C *)
cAngle=PolygonAngle[abcPoly/.φωRep,abc3DVerts[[3]]/.φωRep];
N[%/°/.slRep]
DMSList@%

abcAngles={aAngle,bAngle,cAngle};
abcAnglesN=N[abcAngles/°/.φωRep/.slRep]
Total@%

WP R  & S scale factor(s)

(* As in WP Converted for numerical calculation using my φ *)

wpOrigRscale=1+(Sqrt[((18-11φ)/30)])/.φ->1/φ/.φωRep
N@%
1/%

(* As in WP *)
wpCurrentRscale=1+(Sqrt[((123-76φ+(Sqrt[171941-106264φ]))/(334+16φ))])/.φ->1/φ/.φωRep
N@%
1/%

(* Suggested new R for WP Disdyakis Triacontahedron Article *)
newRscale=(5/(φ+3))/.φ->1/φ/.φωRep/.slRep
(* Converted for numerical calculation using my φ *)
N[%/.φωRep]
((% iNorm)/(φ ))/.φωRep/.slRep

(* Latest new R in WP R *)
wpNewRscale=((5φ )/3)iNorm/.slRep;
FullSimplify[%/.φωRep]
N@%

(* Latest new S in WP R *)
wpNewSscale=((7/φ-6)/(11 iNorm))/.slRep;
FullSimplify[%/.φωRep]
N@%

Document the DMC Cooey Disdyakis Triacontahedron coordinates

(* DMC Cooey Octahedron part of Icosidodecahedron (6) *)
CC8=FullSimplify[3(5+4 (Sqrt[5]))/11/.slRep]
N@%

(* wpNewSscale is the WP Octahedron part of Icosidodecahedron scale *)
DMCfactor=FullSimplify[wpNewSscale/CC8/.φωRep]
N@%

(* DMC Cooey  *)
CC1=(5-(Sqrt[5]))/2;
N@%
FullSimplify[DMCfactor %%/.φωRep]
N@%

(* DMC Cooey *)
CC7=(5+(Sqrt[5]))/2;
N@%
FullSimplify[DMCfactor %%/.φωRep]
N@%

(* DMC Cooey Icosahedron part of an Dodecahedron (12) *)
FullSimplify[DMCfactor Norm[{0,CC1,CC7}]]
N@%

(* DMC Cooey *)
CC3=FullSimplify[3(5+(Sqrt[5]))/10];
N@%
ToRadicals@FullSimplify[DMCfactor %%/.φωRep]
N@%

(* DMC Cooey  *)
CC6=(15+9(Sqrt[5]))/10;
N@%
FullSimplify[DMCfactor %%/.φωRep]
N@%

(* DMC Cooey Unit Icosidodecahedron (12) *)
FullSimplify[DMCfactor Norm[{0,CC3,CC6}]]
N@%

(* DMC Cooey *)
CC0=3(15+(Sqrt[5]))/44;
N@%
FullSimplify[DMCfactor %%/.φωRep]
N@%

(* DMC Cooey *)
CC2=3(5+4(Sqrt[5]))/22;
N@%
FullSimplify[DMCfactor %%/.φωRep]
N@%

(* DMC Cooey *)
CC5=(75+27(Sqrt[5]))/44;
N@%
FullSimplify[DMCfactor %%/.φωRep]
N@%

(* DMC Cooey Icosidodecahedron (30) *)
FullSimplify[DMCfactor Norm[{CC0,CC2,CC5}]]
N@%

(* DMC Cooey *)
CC4=(Sqrt[5]);
N@%
FullSimplify[DMCfactor %%/.φωRep]
N@%

(* DMC Cooey Cube part of an Dodecahedron (8) *)
FullSimplify[DMCfactor Norm[{CC4,CC4,CC4}]]
N@%
1/%

(* DMC Cooey Unit Icosidodecahedron (12) *)
(* DMC Cooey S scale Icosidodecahedron (30) *)
(* DMC Cooey (Sqrt[(5/3) (5-2 (Sqrt[5]))]) =0.9379623704253993` Dodecahedron (12) *)

Visualize of the  Disdyakis Triacontahedron 
    with Icosahedron circumradius=1, (bNorm/aNorm)=.9378 scaled Dodecahedron, S=aNorm=.92101 scaled Icosadodecahedron

DisdyakisTriacontahedronRS=FullSimplify[Join[{
(* 62=32+30=12+20+30 Norms *)

(* Unit scaled Icosahedron (12) **)
(* Scales 10 valent  **)
{"H3",2iNorm{0,0,1}},

(* Dodecahedron (20=Icosahedron (12) + Cube (8)) **)
(* Scales 6 valent **)
(* Icosahedron (12) *)
{"H3",(bNorm/aNorm)(2/(Sqrt[3])){σ,0,0}}/.slRep,
(* Cube (8) *)
{"H3",(bNorm/aNorm) (2/(Sqrt[3])){1,-1,1},"None"},

(* Icosidodecahedron (30=Octahedron (6)+Rhombicuboctahedron (24)) **)
(* Scales 4 valent **)
(* Octahedron (6) **)
{"H3",wpNewSscale {0,σ,0}},
(* Rhombicuboctahedron (24) **)
{"H3",wpNewSscale {-1,τ,σ}}}]/.slRep,Assumptions->φωAssumptions]
dtVertexListRS=oScaleListDo@%;
checkVertices[%,True,True,True,True,False,False]
opacity=.5;
hulls3D@%%%
dtPolyRS=overallHull;

Visualize the new RS scaled (and DMC Cooey) 
   with Icosahedron circumradius=1,  R ((Sqrt[3])wpNewRscale= (bNorm/aNorm)=.9378) scaled Dodecahedron & S=aNorm=.92101 scaled Icosadodecahedron

DisdyakisTriacontahedronRS=FullSimplify[Join[{
(* 62=32+30=12+20+30 Norms *)

(* Unit scaled Icosahedron (12) **)
(* Scales 10 valent  **)
{"H3",2iNorm{0,0,1}},

(* Dodecahedron (20=Icosahedron (12) + Cube (8)) **)
(* Scales 6 valent **)
(* Icosahedron (12) *)
{"H3",(2/(Sqrt[3]))((Sqrt[3])wpNewRscale){σ,0,0}}/.slRep,
(* Cube (8) *)
{"H3", (2/(Sqrt[3]))((Sqrt[3])wpNewRscale){1,-1,1},"None"},

(* Icosidodecahedron (30=Octahedron (6)+Rhombicuboctahedron (24)) **)
(* Scales 4 valent **)
(* Octahedron (6) **)
{"H3",wpNewSscale {0,σ,0}},
(* Rhombicuboctahedron (24) **)
{"H3",wpNewSscale {-1,τ,σ}}}]/.slRep,Assumptions->φωAssumptions]
dtVertexListRS=oScaleListDo@%;
checkVertices[%,True,True,True,True,False,False]
opacity=.5;
hulls3D@%%%
dtPolyRS=overallHull;

2D Polygon angles for the ABC surface triangle for dtVertexListRS

(* Project the face triangle to the XY plane *)
Off[Nearest::mepcmp]
(* Get the index to the 3 vertices A,B,C *)
abc3D=Flatten[FullSimplify[Nearest[dtVertexListRS/.φωRep,#]&/@(oct2ImQuat@#&/@{A1,B1,C1}/.φωRep)],1];
%//MatrixForm
abc3Dindex=position[rndMat[dtVertexListRS/.φωRep],#]&/@rndMat@abc3D

(* Define a 3D Polygon from the ABC vertices *)
abc3DVerts=FullSimplify[dtVertexListRS[[abc3Dindex]],Assumptions->φωAssumptions];
%//MatrixForm
%/.φωRep//MatrixForm
N@%//MatrixForm

(* Get the 3D edge lengths for AB, AC, BC *)
FullSimplify[Norm@{abc3DVerts[[#[[1]]]]-abc3DVerts[[#[[2]]]]}/.slRep,Assumptions->φωAssumptions]&/@Subsets[Range@3,{2}];
%/.φωRep//MatrixForm
N@%//MatrixForm

(* Define a MTM Polygon *)
abcPoly=Polygon@abc3DVerts;

(* Get angle A *)
aAngle=PolygonAngle[abcPoly/.φωRep,abc3DVerts[[1]]/.φωRep];
N[%/°/.slRep]
DMSList@%

(* Get angle B *)
bAngle=PolygonAngle[abcPoly/.φωRep,abc3DVerts[[2]]/.φωRep];
N[%/°/.slRep]
DMSList@%

(* Get angle C *)
cAngle=PolygonAngle[abcPoly/.φωRep,abc3DVerts[[3]]/.φωRep];
N[%/°/.slRep]
DMSList@%

abcAngles={aAngle,bAngle,cAngle};
abcAnglesN=N[abcAngles/°/.φωRep/.slRep]
Total@%

Angles of AB, BC, and AC projected to the XY plane (changing the triangle shape)

abcPrj={{1,0,0},{0,1,0},{0,0,0}};
abc3DPrj=#.abcPrj&/@abc3DVerts;
%//MatrixForm

abc2DPrj=%[[All,;;2]];
FullSimplify[%/.φωRep,Assumptions->φωAssumptions]//MatrixForm
N@%//MatrixForm

(* Get the 3D edge lengths for AB, AC, BC
Notice they are different than the unprojected ! *)
FullSimplify[Norm@{abc3DPrj[[#[[1]]]]-abc3DPrj[[#[[2]]]]}/.slRep,Assumptions->φωAssumptions]&/@Subsets[Range@3,{2}]
%/.φωRep//MatrixForm
N@%//MatrixForm

abPrj=VectorAngle[abc2DPrj[[1]],abc2DPrj[[2]]]
%/.φωRep//MatrixForm
N@%//MatrixForm
%/°
DMSList@%

bcPrj=FullSimplify[VectorAngle[abc2DPrj[[2]],abc2DPrj[[3]]],Assumptions->φωAssumptions]
%/.φωRep//MatrixForm
N@%//MatrixForm
%/°
DMSList@%

acPrj=FullSimplify[VectorAngle[abc2DPrj[[1]],abc2DPrj[[3]]],Assumptions->φωAssumptions]
%/.φωRep//MatrixForm
N@%//MatrixForm
%/°
DMSList@%

Total@N[{abPrj,acPrj,bcPrj}/°/.φωRep/.slRep]

Visualize the Great Rhombicosidodecahedron with circumradius=1

(* Set the Unit Norm for the Λ(111) Great Rhombicosidodecahedron (120 regular& proper aka.Truncated icosidodecahedron)*)
FullSimplify[σ{α0,β0,γ0}/.aRule@{1,1,1}/.φωRep]
FullSimplify[σ(({α0,β0,γ0}/.aRule@{1,0,0})+({α0,β0,γ0}/.aRule@{0,1,0})+{α0,β0,γ0}/.aRule@{0,0,1})/.φωRep]
%%==%

αβγ111UnitNorm=FullSimplify[1/Norm@%%]
N@%
1/%

GreatRhombicosidodecahedron=FullSimplify[{
{{σ,σ,τ+3}/.slRep},
{{3 ,τ,2τ}/.slRep},
{{2 ,τ-σ,τ+2}/.slRep},
{{ 2σ,(τ^3),τ}/.slRep},
{{σ,3τ-1,(τ^2)}/.slRep}},Assumptions->φωAssumptions]

grVertsNoScaling=oScaleListDo[GreatRhombicosidodecahedron[[;;5]]]/.slRep;
grVerts=grVertsNoScaling αβγ111UnitNorm/2 /.slRep;

(*)
checkVertices[grVerts,False,True,True,True,True,False]

opacity=.1;
hulls3D@grVerts
grPoly=overallHull;

Get the 10 valent verts ,  create a Polygon and get the centroid which must be a Disdyakis Triacontahedron vertex

N[(1/(2η1))((oct2ImQuat@A1)/bNorm)/.φωRep/.slRep];
%//MatrixForm

gr10Nearest=FullSimplify[Nearest[grVerts/.φωRep,%%,10]];
N@%//MatrixForm

gr10NearestIndex=position[rndMat[grVerts/.φωRep],#]&/@rndMat@%

gr10NearestSymbolic=grVerts[[%]];
%//MatrixForm

gr10Poly=Polygon@%%;
gr10Centeroid=N@RegionCentroid[%/.φωRep]

Get the 6 valent verts,  create a Polygon and get the centroid which must be a Disdyakis Triacontahedron vertex

N[(**).965(**) ((oct2ImQuat@B1)/bNorm)/.φωRep/.slRep];
%//MatrixForm

gr6Nearest=FullSimplify[Nearest[grVerts/.φωRep,%%,6]];
N@%//MatrixForm

gr6NearestIndex=position[rndMat[grVerts/.φωRep],#]&/@rndMat@%
gr6NearestSymbolic=grVerts[[%]];
%//MatrixForm

gr6Poly=Polygon@%%;
gr6Centeroid=Chop@N@RegionCentroid[%/.φωRep]

Get the 4 valent verts,  create a Polygon and get the centroid which must be a Disdyakis Triacontahedron vertex

N[((oct2ImQuat@C1)/(bNorm cNorm))/.φωRep/.slRep];
%//MatrixForm

gr4Nearest=FullSimplify[Nearest[grVerts/.φωRep,%%,4]];
N@%//MatrixForm

gr4NearestIndex=position[rndMat[grVerts/.φωRep],#]&/@rndMat@%
gr4NearestSymbolic=grVerts[[%]];
%//MatrixForm

gr4Poly=Polygon@%%;
gr4Centeroid=N@RegionCentroid[%/.φωRep]

Create the Disdyakis Triacontahedron ABC Polygon from the  centroids

grCentroids={gr10Centeroid,gr6Centeroid,gr4Centeroid};
%//MatrixForm

(* Define a MTM Polygon *)
abcPoly=Polygon@grCentroids;

(* Get angle A *)
aAngle=PolygonAngle[abcPoly,grCentroids[[1]]]
FullSimplify[%/.φωRep]
N@%
%/°
DMSList@%

(* Get angle B *)
bAngle=PolygonAngle[abcPoly,grCentroids[[2]]]
FullSimplify[%/.φωRep]
N@%
%/°
DMSList@%

(* Get angle C *)
cAngle=PolygonAngle[abcPoly,grCentroids[[3]]]
FullSimplify[%/.φωRep]
N@%
%/°
DMSList@%

abcAngles={aAngle,bAngle,cAngle};
abcAnglesN=N[abcAngles/°/.φωRep/.slRep]
Total@%

Verify the ABC angles against the three equations linking each plane angle to the dihedral

Sin[θ/2]=((Cos[π/p])/(Cos[(Subscript[α, p])/2])),
Sin[θ/2]=((Cos[π/q])/(Cos[(Subscript[α, q])/2])),
Sin[θ/2]=((Cos[π/r])/(Cos[(Subscript[α, r])/2])),
Where p, q, r are 4, 6, and 10 in some order, θ is the dihedral, and the fourth equation is that the sum of the three alphas should be 180 degrees.

(* Sin of the Dihedral half angle *)
N[Sin[θ/2]]/°

((Cos[π/#[[1]]])/(Cos[#[[2]]/2]))/°&/@N[{{10,aAngle},{6,bAngle},{4,cAngle}}/.φωRep/.slRep]